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Introduction to Angles
Latressa T R Cobb
RAY: A ray, AB, starts at point A and
goes on forever in one direction.
ANGLE: An angle is formed by two
rays with a common endpoint.
A
B
A
B
VERTEX: A vertex is the point where two or more
rays (or segments in a polygon) intersect.
Angles can be named several ways . . .
C
A
B
1 means
1
A means
A
C
ABC means
B
C
This 3rd method is used when more than two rays form a vertex.
D
Which is B?
C
B
A
This would be too confusing!
Therefore we use:
ABC, ABD or CBD
When we want to indicate that two or more segments or
angles are equal in a figure, we use an equal number of
slash marks like the figures below.
We can classify the different types of angles.
Acute Angle: Any angle with a measure between
(but not including) 0º and 90º.
Right Angle: Any angle which measures 90º.
Obtuse Angle: Any angle with a measure between
(but not including) 90º and 180º.
Straight Angle: Straight Angles (or lines) have a
measure of 180º.
We will classify some of the different RELATIONSHIPS of angles.
Supplementary Angles:
Any pair of angles whose sum is 180º.
D
160º
30º
A
B
A
O
20º
B
LINEAR PAIR: A supplementary pair of angles
that form a straight angle.
Complementary Angles:
Any pair of angles whose sum is 90º.
30º
50º
40º
60º
Parallel lines: lines that are in the same plane
and do not intersect (cross).
Transversal: a line that intersects two or more
lines in the same plane.
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When we classify angle relationships, we need to look at the
POSITION of the angles in relation to the parallel lines AND
the transversal.
Interior means the angles are
between the parallel lines.
Exterior means the angles are
outside the parallel lines.
Two angles can be on the same side of the transversal
or they can be on alternate sides of the transversal.
Same side
Alternate sides
Now, we put these two ideas together to classify angle relationships.
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Angle 4 & 5 are on alternating sides of the transversal,
and are interior of the parallel lines.
So, we say that angle 4 &
angle 5 are Alternate Interior
Angles.
2
1
3
4
6
5
8
7
**Note when we are classifying angles on parallel lines, we are dealing with PAIRS
of angles classified by their positions in relation to two things: the parallel lines
AND the transversal. We do NOT say that 8 is an exterior angle (this will be
defined later as a single classification).
The angle relationships when parallel lines are cut by a transversal are…
Alternate Interior Angles
alternate sides of the transversal and interior of the parallels
Same Side Interior Angles
Same side of the transversal and interior of the parallels
Alternate Exterior Angles
alternate sides of the transversal and exterior of the parallels
Same Side Exterior Angles
Same side of the transversal and exterior of the parallels
There is one other important type of angle relationship on parallel lines
cut by a transversal but it doesn’t quite follow the same ideas as the
first four…
Corresponding Angles
Angles on the same side of the transversal and in the
same position on the parallel lines (either the angles are
both above or both below the parallel lines.)
Since the five types of angles describe angle relationships, we can
only classify PAIRS of angles as Alternate Interior Angles, Same
Side Interior Angles, Alternate Exterior Angles, Same Side
Exterior Angles, or Corresponding Angles.
Pairs of angles can be classified as one of these relationships even
if the lines being cut by the transversal are NOT parallel.
Find pairs of angles that are:
Same side interior angles
(3 & 5) or (4 & 6)
2
1
4
3
5
6
7
Alternate interior angles
(3 & 6) or (4 & 5)
Alternate exterior angles
(1 & 8) or (2 & 7)
Same side exterior angles
(2 & 8) or (1 & 7)
Corresponding angles:
(1 & 5) or (2 & 6)
or (3 & 7) or (4 & 8)
8
If (and only if) the lines cut by the transversal are PARALLEL, then we can make
some conclusions based on the angle relationships.
Some pairs are equal and some are supplementary.
b
a
c
d
e
f
h
g
So, if the lines are parallel, what can you conclude about:
Alternate Interior Angles
Same Side Interior Angles
Alternate Exterior Angles
Same Side Exterior Angles
Corresponding Angles
Equal
Supplementary
Equal
Supplementary
Equal
Vertical Angles:
The opposite angles formed by two
intersecting lines. This is a relationship
between PAIRS of angles…one angle cannot
be vertical.
Conjecture: When two lines intersect, the resulting
equal
pairs of vertical angles are ________.
Adjacent Angles:
Angles that are “next to” each other.
To be adjacent angles, they MUST
satisfy 3 conditions:
 share a common side,
 share a common vertex,
 No overlap … the common side is IN
BETWEEN the two angles.